On the two smallest Pisot numbers. (English. Russian original) Zbl 1284.11106

Math. Notes 94, No. 5, 820-823 (2013); translation from Mat. Zametki 94, No. 5, 784-787 (2013).
Summary: Recall that a Pisot number is an algebraic integer greater than \(1\), whose conjugates lie strictly inside the unit disc \(\mathbb{D} = \{ z \in \mathbb{C}: |z| < 1 \}\). Pisot numbers can be characterized by their diophantine properties. Consider
\[ L(\theta): \sup_{\xi \in \mathbb{R}}\liminf_{n \to \infty} \| \xi \theta^n \| \] . The author proves the following theorem: let \(\theta\) be the greatest root of the equation \(x^3-x-1=0\), then the equality \(L(\theta)= 1/5\) holds. Similarly, she proves that if \(\theta\) is the greatest root of the equation \(x^4-x^3-1=0\), then the equality \(L(\theta)=3/17\) holds.


11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11J25 Diophantine inequalities
Full Text: DOI


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